Sum of Lognormal Random Variables
A variable X is lognormally distributed when the natural logarithm Y=ln(X) is normally distributed. In many physical and financial analysis systems lognormal distributions are ubiquitous. Lognormal distributions can also be used for modeling mineral resources, pollutants, chemical sensitivities, etc., Crow et al. (Eds.), Lognormal Distributions: Theory and Application, Dekker, New York, 1988.
In another application, the attenuation due to shadowing in wireless channels is often modeled by a lognormal distribution. Hence, in the analysis of wireless systems, one often encounters the sum of lognormal random variables (RV). For example, the sum of lognormal RVs characterizes the total co-channel interference (CCI) at a receiver from all the transmissions in neighboring wireless cells. For brevity, the distribution of the sum of lognormal RVs can be referred to as the lognormal sum distribution.
The lognormal distribution is also of interest in outage probability analysis, G. L. Stüber, Principles of Mobile Communications, Kluwer Academic Publishers, 1996; and in ultra wide band systems, H. Liu, “Error performance of a pulse amplitude and position modulated ultra-wideband system over lognormal fading channels,” IEEE Commun. Lett., vol. 7, pp. 531-533, 2003.
Given the importance of the lognormal sum distribution in wireless communications, as well as in other fields such as optics and reliability theory, considerable efforts have been devoted to analyzing its statistical properties. While exact closed-form expressions for the lognormal sum probability distribution functions (PDFs) are unknown, several analytical approximation methods are known, L. F. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,” IRE Trans. Commun. Syst., vol. CS-8, pp. 57-67, 1960; S. Schwartz and Y. Yeh, “On the distribution function and moments of power sums with lognormal components,” Bell Syst. Tech. J., vol. 61, pp. 1441-1462, 1982; N. C. Beaulieu and Q. Xie, “An optimal lognormal approximation to lognormal sum distributions,” IEEE Trans. Veh. Technol., vol. 53, pp. 479-489, 2004; S. B. Slimane, “Bounds on the distribution of a sum of independent lognormal random variables,” IEEE Trans. Commun., vol. 49, pp. 975-978, 2001; D. C. Schleher, “Generalized gram-charlier series with application to the sum of log-normal variates,” IEEE Trans. Inform. Theory, pp. 275-280, 1977; and F. Berggren and S. Slimane, “A simple bound on the outage probability with lognormally distributed interferers,” IEEE Commun. Lett., vol. 8, pp. 271-273, 2004.
The prior art methods can be classified broadly into two categories, those that use a single distribution and those that use compound distributions. The methods by Fenton-Wilkinson (F-W), Schwartz-Yeh (S-Y), and Beaulieu-Xie (B-X) approximate the lognormal sum by a single lognormal RV. The proven permanence of the lognormal PDF when the number of summands becomes very large lends further credence to those methods, W. A. Janos, “Tail of the distributions of sums of lognormal variates,” IEEE Trans. Inform. Theory, vol. IT-16, pp. 299-302, 1970; and R. Barakat, “Sums of independent lognormally distributed random variables,” J. Opt. Soc. Am., vol. 66, pp. 211-216, 1976.
The methods by Farley, Ben Slimane, and Schleher instead compute a compound distribution based on the properties of the lognormal RV. The compound distribution can be specified in several ways. For example, the methods in S. Schwartz and Y. Yeh, “On the distribution function and moments of power sums with lognormal components,” Bell Syst. Tech. J., vol. 61, pp. 1441-1462, 1982 and S. B. Slimane, “Bounds on the distribution of a sum of independent lognormal random variables,” IEEE Trans. Commun., vol. 49, pp. 975-978, 2001, specify the approximating distribution in terms of strict lower bounds of the cumulative distribution function (CDF), while D. C. Schleher, “Generalized gram-charlier series with application to the sum of log-normal variates,” IEEE Trans. Inform. Theory, pp. 275-280, 1977, partitions the range of the lognormal sum into three segments, with each segment being approximated by a distinct lognormal RV.
Beaulieu et al. describe the accuracy of several of the above methods, and have shown that all the methods have their own advantages and disadvantages; none is unquestionably better than the others. The F-W method is inaccurate for estimating the CDF for small values of the argument, while the S-Y method is inaccurate for estimating the complementary CDF (CCDF) for large values of the argument. The Farley's method and, more generally, the formulae derived in S. B. Slimane, “Bounds on the distribution of a sum of independent lognormal random variables,” IEEE Trans. Commun., vol. 49, pp. 975-978, 2001, are strict bounds that can be loose approximations for certain typical parameters of interest. The methods also differ considerably in their complexity. For example, the S-Y method involves solving non-linear equations and requires an iterative procedure to handle the sum of more than two RVs. Only the F-W method offers a closed-form solution for calculating the underlying parameters of the approximating lognormal PDF.
Moment Generating Function Based Methods
The desirable property of a moment generating function (MGF) and the characteristic function (CF)—that the MGF (or CF) of a sum of independent RVs can be written as the product of the MGFs (or CFs) of the individual RVs, A. Papoulis, Probability, Random Variables and Stochastic Processes. McGraw Hill, 3rd ed., 1991—is well known. However, the methods that use this property to approximate the distribution of the sum of lognormal RVs by a lognormal distribution, generally require an extremely accurate numerical computation at a sufficiently large number of sample points. In addition, the methods are relatively complex, N. C. Beaulieu and Q. Xie, “An optimal lognormal approximation to lognormal sum distributions,” IEEE Trans. Veh. Technol., vol. 53, pp. 479-489, 2004; and R. Barakat, “Sums of independent lognormally distributed random variables,” J. Opt. Soc. Am., vol. 66, pp. 211-216, 1976. While the CF is a special case of the MGF, the two are treated separately for clarity of the following description. Moreover, the MGF-based methods have only considered the case where the lognormal RVs are independent of each other. The case for correlated RVs has not been explored.
Barakat first determined the CF of the lognormal distribution using a Taylor series expansion, and then applied an inverse Fourier transform to the product of the lognormal CFs to determine the PDF of the lognormal sum of RVS. However, the oscillatory property of the Fourier integrand as well as the slow decay rate of the lognormal PDF tail makes the numerical evaluation difficult and inaccurate. Also, no effort has been made to find the analytical expressions of the approximate distribution from the numerically computed PDF of the sum.
A similar method was described by H. R. Anderson, “Signal-to-interference ratio statistics for am broadcast groundwave and skywave signals in the presence of multiple skywave interferers,” IEEE Trans. Broadcasting, vol. 34, pp. 323-330, 1988. The Beaulieu-Xie's method first numerically evaluated the lognormal sum CDF at several sample points with very high accuracy using a modified Clenshaw-Curtis method. The composite CDF is obtained by numerically calculating the inverse Fourier transform of the lognormal sum. The CDF is plotted on ‘lognormal paper’, in which the lognormal PDF appears as a straight line. The parameters of the approximating lognormal distribution are determined by minimizing the maximum (minimax) error in a given interval. While that method is optimal in the minimax sense on lognormal paper, this does not imply optimality in directly matching the probability distribution.
Sum of Correlated Lognormal Random Variables
The correlated case has been investigated in the literature. Extensions to the F-W method have been proposed in A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities in the presence of correlated lognormal interferers,” IEEE Trans. Veh. Technol., vol. 43, pp. 164-173, 1994; and F. Graziosi, L. Fuciarelli, and F. Santucci, “Second order statistics of the SIR for cellular mobile networks in the presence of correlated co-channel interferers,” in Proc. VTC, pp. 2499-2503, 2001. Extension to the S-Y method to handle the correlated case was proposed in A. Safak, “Statistical analysis of the power sum of multiple correlated log-normal components,” IEEE Trans. Veh. Technol., vol. 42, pp. 58-61, 1993; extensions to the Cumulants method have been considered in A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities in the presence of correlated lognormal interferers,” IEEE Trans. Veh. Technol., vol. 43, pp. 164-173, 1994. But Farley's method, the Beaulieu-Xie method, and the bounds of Ben-Slimane et al. do not apply to the sum of correlated lognormal RVs. Outage probability bounds, which, in effect, specify a compound distribution, are derived in F. Berggren and S. Slimane, “A simple bound on the outage probability with lognormally distributed interferers,” IEEE Commun. Lett., vol. 8, pp. 271-273, 2004 using the arithmetic-geometric mean inequality and can handle the correlated case.
However, the basic limitations of the various methods still apply: the S-Y extension cannot accurately estimate small values of the CCDF, the F-W extension again cannot accurately estimate small values of the CDF, and the bounds are loose for larger logarithmic variances.
Sum of Suzuki or Lognormal-Rice Random Variables
An extension of the F-W-based moment matching method to approximate the distribution of a sum of Suzuki RVs by a lognormal distribution is described by F. Graziosi and F. Santucci, “On SIR fade statistics in Rayleigh-lognormal channels,” Proc. ICC, pp. 1352-1357, 2002. Another technique is a two-step approximation process in which each of the lognormal-Rician or Suzuki RVs is first approximated by a lognormal RV, by equating the means and variances, and then the sum of the lognormal RVs is again approximated by a single lognormal RV using the F-W or the S-Y methods.
The sum of Suzuki RVs has also been approximated by another Suzuki RV, J. E. Tighe and T. T. Ha, “On the sum of multiplicative chi-square-lognormal random variables,” Proc. Globecom, pp. 3719-3722, 2001. Exact formulae are known that express the outage probability of a sum of lognormal-Rician RVs in the form of a single integral, which is evaluated numerically, J.-P. M. Linnartz, “Exact analysis of the outage probability in multiple-user radio,” IEEE J. Select. Areas Commun., vol. 10, pp. 20-23, 1992; and M. D. Austin and G. L. Stuber, “Exact co-channel interference analysis for log-normal shadowed Rician fading channels,” Electron. Lett., vol. 30, pp. 748-749, 1994.
However, the problem of approximating any linear combination of lognormal random variables with a single lognormal RV was not addressed.